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Elementary calculus (differentiation) is used to obtain information on a line which touches a curve at one point (i.e. a tangent). This is done by calculating the gradient, or slope of the curve, at a single point. However, this does not provide us with reliable information on the curve's actual value at given points in a wider interval. This is where the concept of power series becomes useful.

Consider the curve of y=cos⁡(x){\displaystyle y=\cos(x)} , about the point x=0{\displaystyle x=0} . A naïve approximation would be the line y=1{\displaystyle y=1} . However, for a more accurate approximation, observe that cos⁡(x){\displaystyle \cos(x)} looks like an inverted parabola around x=0{\displaystyle x=0} - therefore, we might think about which parabola could approximate the shape of cos⁡(x){\displaystyle \cos(x)} near this point. This curve might well come to mind:

y=1−x22{\displaystyle y=1-{\frac {x^{2}}{2}}}

In fact, this is the best estimate for cos⁡(x){\displaystyle \cos(x)} which uses polynomials of degree 2 (i.e. a highest term of x2{\displaystyle x^{2}}) - but how do we know this is true? This is the study of power series: finding optimal approximations to functions using polynomials.

this converges when |x|<1{\displaystyle |x|<1} , the range f(x)−1<x<1{\displaystyle f(x)-1<x<1} , so the radius of convergence - centered at 0 - is 1. It should also be observed that at the extremities of the radius, that is where x=1{\displaystyle x=1} and x=−1{\displaystyle x=-1} , the power series does not converge.

which is always true - therefore, this power series has an infinite radius of convergence. In effect, this means that the power series can always be used as a valid alternative to the original function, ex{\displaystyle e^{x}} .

which will also converge for |x|<1{\displaystyle |x|<1} . When x=−1{\displaystyle x=-1} this is the harmonic series, which diverges; when x=1{\displaystyle x=1} this is an alternating series with diminishing terms, which converges to ln⁡(2){\displaystyle \ln(2)} - this is testing the extremities.

It also lets us write series for integrals we cannot do exactly such as the error function: